Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly internal to some symmetric monoidal category).

What if we add defects into the game? I couldn't find a hard classification result for 2d extended TQFTs with defects. Is there one? Or at least a folk theorem? Does the classification of extended TQFTs without defects somehow follow as a special case?

EDIT: If there are hints on how the classification for non-extended 2d TQFTs (2-1-TQFTs, so to speak) with defects goes, that would be relevant too.

  • $\begingroup$ Does "with defects" mean something different than "with singularities"? $\endgroup$ – Qiaochu Yuan Apr 30 '15 at 16:54
  • $\begingroup$ @QiaochuYuan, I think this is regarded to be the same. For simplicity, we could leave point defects out for the time being and just ask for TQFTs with line defects. Then the lines should just be submanifolds. $\endgroup$ – Manuel Bärenz Apr 30 '15 at 17:02
  • $\begingroup$ @QiaochuYuan, thinking about it again, defects are probably more than just singularities. Looking at arxiv.org/pdf/1107.0495v1.pdf, it seems that defects should actually give morphisms of TQFTs. $\endgroup$ – Manuel Bärenz Oct 30 '15 at 12:02

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